2.1 Basic Concepts and Formulae 89
whereμis the reduced mass given by
μ=memp
me+mp(2.17)
andZ=1 for H-atom.
The radius of the smallest orbit, called the Bohr radius,
a 0 =r 1 =ε 0 h^2
πme^2= 0. 529 A ̊ (2.18)
Orbital velocity in thenth orbit
vn=ze^2
2 εonh(2.19)
Fine structure constant
α=v
c=
e^2
2 εohc=
1
137
(2.20)
αis a measure of the electromagnetic interaction
Kinetic Energy of electron
Kn=1
2
mvn^2 =ze^2
8 πεorn(2.21)
Potential energy of electron
Un=−ze^2
4 πεorn(2.22)
Total Energy of electron
En=Kn+Un=−ze^2
8 πεorn=−
mz^2 e^4
8 ε^2 oh^2 n^2=
− 13. 6
n^2(ev) (2.23)ν ̃if=1
λif=R
(
1
n^2 f−
1
n^2 i)
(2.24)
Mesic atom
A negatively charged muon or pion when captured by the nucleus forms a bound
system called mesic atom before absorption or decay system.
i. rn ∝ 1 /μ; therefore radii are shrunk by a factor of∼200 for muonic atom
compared to H-atom.
ii. En ∝ μ; therefore energy levels are spaced 200 times greater than those of
H-atom. X-rays are emitted instead of visible light, when the muon cascades
down.